Paul Dirac and Matrix mechanics

Question

Did Paul Dirac in 1925 derive the Heisenberg's matrix mechanics from the Newtonian mechanics and canonical commutation relation? That is, from the expressions $$P=m\frac{dX}{dt}, \space \space \space \space \space \space -\nabla V=\frac{dP}{dt} \space \space \space \space \space \space \text{and} \space \space \space \space \space \space [X,P]=i\hbar I.$$

If the derivation is possible, how it was done? In a book Lectures on Quantum Mechanics 2ed by Basdevant on pages 185-186, we have the following story:

Before we start, let us say a few words about how Dirac entered quantum mechanics in 1925. Dirac was a student in Cambridge. He was born on August 8, 1902 so he was barely 23. One can talk for hours about him. He was completely unusual. Very discreet, very polite, and careful, he thought a lot. He had a great culture, in particular in mathematics. He knew about noncommutative algebras, but let’s not anticipate. At that time, Cambridge had an impressive intellectual richness and was similar to Göttingen in that respect. The professors were Keynes in economics, J.J. Thomson, E. Rutherford, A.S. Eddington, and Fowler in physics, as well as others. And there, by accident, everything got started. On July 28, 1925, Heisenberg was invited to give a talk in Cambridge at the very fashionable Kapitza club. That talk had no influence on Dirac (who kept thinking instead of listening); neither of them remembered seeing the other on that day. But Heisenberg gave his article to Fowler who gave it to Dirac. He read the paper with some difficulty: it was full of philosophical ideas, but the mathematical formalism was clumsy. Dirac had nothing against philosophy, but he didn’t particularly care for it in that context; what he wanted was good mathematics. Very rapidly he said, “This will lead us nowhere.” Two weeks later, he walked into Fowler’s office and said, “It’s remarkable; it contains the key of quantum mechanics!” What had happened is that Heisenberg had cracked up! Since 1924, he had elaborated a system of calculational rules and a theory that worked well. Then Born had gotten involved in the whole business, and he had found that there were matrices. They had seen Hilbert. And Heisenberg, a positivist, had learned something tragic about his theory: some physical quantities did not commute! The product of a and b was not the same as the product of b and a! And that was contrary to any physical sense. The product of two quantities had never depended on the order. So Heisenberg was terrified, “My theory is not beautiful!” He was so terrified that he had put all the dust under the carpet and he wrote his papers in such a way to hide this noncommutativity as well as he could. Dirac, very meticulously, had redone the calculations step by step, and he realized that everything boiled down to this noncommutativity. In particular, he proved independently from Born and Jordan the fundamental relation $[X,P]=iI$ in August 1925 without mentioning matrices. He knew the existence of noncommutative algebras. So he made an attempt to modify the classical equations in order to take into account this noncommutativity. After all nothing dictates that physical quantities should commute. Some time after, Dirac said, “You know, Heisenberg was scared. He was scared by these foreign mathematics. I can understand that he was scared: all his theory was at stake. I had an enormous advantage over him: I had no risk.” So, Dirac constructed his own version of quantum mechanics, based on noncommutative algebras, that is, on commutators.

Answer 1

Indeed, HSM has world-class experts trained to shed ample light on such things. I slightly object to Jean-Louis B's colorized storytelling version, and especially the imputation that Heisenberg deliberately obscured noncommutativity; it is true he did not emphasize it because he was not comfortable with matrices, and was out of his element in noncommutative structures. But the alleged Dirac quote does capture their basic asymmetry.

Dirac's epochal 1925 paper does indeed capture the essence of QM, through the basic (Born) relation (12), following from (11), in modern notation, $$ x\star y -y\star x \approx i\hbar \{x,y\}, \\ [\hat x ,\hat y] \mapsto i\hbar \{x,y\} \tag {11} $$ the thoroughly noncommutative Poisson Bracket. Dirac had a background in mathematics, possibly Lie theory, and engineering, both of which converged on this formulation!

(Caution: I'm using the modern symbol in lieu of Dirac's commutator-looking one, the [.,.]! I have also used the approximate sign to suggest that, in modern notation of suitable corresponding commutative variables composed noncommutatively, the two expressions are not equal: the l.h.s. is a Moyal bracket that starts with the PB, but subsequently contains $\hbar$-corrections over and above the PB. Dirac resisted sorting those out and opposed Moyal who did, 20 years after that. I have used the ↦ symbol to denote the quietly ambiguous quantization map, in Hilbert space. But, no matter: the fundamental insight holds and changed the intellectual history of the 20th century.)

A historian of science would have to parse out if $[q,p]=i\hbar$ was indeed introduced by Born$^\natural$ before this insight, or subsequently and independently of it, and whether Dirac was aware of that, or not... Probably not. (At that time, possibly via Jordan, Born had identified the variables as matrices, naturally noncommutative.)

In any case, given the intuitive conversion dictionary, and his proof of the commutator rules and the Jacobi identity, conversion of the PBs,

$$\dot{p}=\{ p, H\},~~\dot{ x}= \{ x,H\} \tag{14},$$ yields the basic commutator expressions for quantum variables, $$i\hbar \dot{\hat p}=[\hat p,\hat H] ,~~i\hbar \dot{\hat x}= [\hat x,\hat H],$$ that students of QM start their courses with. In excessive generosity, Dirac would call these the "Heisenberg ('s form of the) equations of motion" for the rest of his life...

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$^\natural$ Born, Heisenberg and Jordan, 1926, equation (12). Both Dirac's and this manuscript were submitted in November 1925, with an interval of 9 days... It's hard to know who knew what before manuscript submission.

• Dirac, P. A. M. (1925). The fundamental equations of quantum mechanics. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 109 (752), 642-653.

• Born, M., Heisenberg, W., & Jordan, P. (1926). Zur Quantenmechanik. II. Zeitschrift für Physik, 35 (8), 557-615.